We present a possible computational content of the negative translation of classical analysis with the Axiom of Choice. Our interpretation seems computationally more direct than the one based on Godel's Dialectica interpretation [10, 18]. Interestingly, thisinterpretation uses a re nement of the realizibility semantics of the absurdity proposition, which is not interpreted as the empty type here. We alsoshowhow to compute witnesses from proofs in classical analysis, and how to interpret the axiom of Dependent Choice and Spector's Double Negation Shift.

In [15],[16] Kreisel introduced the no-counterexample interpretation (n.c.i.) of Peano arithmetic. In particular he proved, using a complicated "-substitution method (due to W. Ackermann), that for every theorem A (A prenex) of first-order Peano arithmetic PA one can find ordinal recursive functionals \Phi A of order type ! " 0 which realize the Herbrand normal form A of A. Subsequently more

Abstract. Extending Gödel’s Dialectica interpretation, we provide a functional interpretation of classical theories of positive arithmetic inductive definitions, reducing them to theories of finite-type functionals defined using transfinite recursion on well-founded trees. 1.

This article will survey the state of the art nowadays, in particular recent advance in proof theory beyond admissible proof theory, giving some prospects of success of obtaining an ordinal analysis of \Pi

Martin-Lof's type theory can be described as an intuitionistic theory of iterated inductive definitions developed in a framework of dependent types. It was originally intended to be a full-scale system for the formalization of constructive mathematics, but has also proved to be a powerful framework for programming. The theory integrates an expressive specification language (its type system) and a functional programming language (where all programs terminate). There now exist several proof-assistants based on type theory, and many non-trivial examples from programming, computer science, logic, and mathematics have been implemented using these. In this series of lectures we shall describe type theory as a theory of inductive definitions. We emphasize its open nature: much like in a standard functional language such as ML or Haskell the user can add new types whenever there is a need for them. We discuss the syntax and semantics of the theory. Moreover, we present some examples ...

In this paper we shall investigate fragments of Kripke--Platek set theory with Infinity which arise from the full theory by restricting Foundation to \Pi n Foundation, where n 2. The strength of such fragments will be characterized in terms of the smallest ordinal ff such that L ff is a model of every \Pi 2 sentence which is provable in the theory. 1 Introduction Kripke-Platek set theory plus Infinity (hereinafter called KP!) is a truly remarkable subsystem of ZF. Though considerably weaker than ZF, a great deal of set theory requires only the axioms of this subsystem (cf.[Ba]). KP! consists of the axioms Extensionality, Pair, Union, (Set)Foundation, Infinity, along with the schemas of \Delta 0 --Collection, \Delta 0 --Separation, and Foundation for Definable Classes. So KP! arises from ZF by completely omitting Power Set and restricting Separation and Collection to \Delta 0 --formulas. These alterations are suggested by the informal notion of "predicative". KP! is an impredicat...

this article has appeared in Computational Logic and Proof Theory (Proc. 3

Abstract. We prove that the (non-intuitionistic) law of the double negation shift has a bounded functional interpretation with bar recursive functionals of finite type. As an application, we show that full numerical comprehension is compatible with the uniformities introduced by the characteristic principles of the bounded functional interpretation for the classical case. §1. Introduction and background. In 1962 [14], Clifford Spector gave a remarkable characterization of the provably recursive functionals of full secondorder arithmetic (a.k.a. analysis). The central result of his paper is an extension, from arithmetic to analysis, of the (then quite recent) dialectica interpretation of Gödel of 1958 [7]. Spector’s extension relies on a form of well-founded recursion

. The paper investigates the strength of the Anti-Foundation Axiom, AFA, on the basis of Kripke-Platek set theory without Foundation. It is shown that the addition of AFA considerably increases the proof theoretic strength. MSC:03F15,03F35 Keywords: Anti-foundation axiom, Kripke-Plate set theory, subsystems of second order arithmeic 1. Introduction Intrinsically circular phenomena have come to the attention of researchers in differing fields such as mathematical logic, computer science, artificial intelligence, linguistics, cognitive science, and philosophy. Logicians first explored set theories whose universe contains what are called non-wellfounded sets, or hypersets (cf. [6], [2]). But the area was considered rather exotic until these theories were put to use in developing rigorous accounts of circular notions in computer science (cf. [4]). Instead of the Foundation Axiom these set theories adopt the so-called AntiFoundation Axiom, AFA, which gives rise to a rich universe of ...

It can be traced back to Brouwer that continuous functions of type StrA → B, where StrA is the type of infinite streams over elements of A, can be represented by well founded, A-branching trees whose leafs are elements of B. This paper generalises the above correspondence to functions defined on final coalgebras for power-series functors on the category of sets and functions. While our main technical contribution is the characterisation of all continuous functions, defined on a final coalgebra and taking values in a discrete space by means of inductive types, a methodological point is that these inductive types are most conveniently formulated in a framework of dependent type theory.